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High Dimensional Expanders, Ramanujan Complexes and Codes

Periodic Reporting for period 4 - HIEXP (High Dimensional Expanders, Ramanujan Complexes and Codes)

Reporting period: 2021-02-01 to 2021-07-31

Expander graphs have been an important subject of research in mathematics and computer science in the last five decades. These graphs have found numerous applications in computer science ( Networks, algorithms, error correcting codes and many more) but also in pure mathematics ( groups, number theory, geometry and more).
The goal of the current project is to develop high dimensional theory of expanders. We already have some applications, but the history of expander graphs teaches us that we should expect also more unexpected applications. indeed, already in the first two years of the project we came to a totally unexpected application to non-approximated groups: this is solving a first case of a series of problems which were completely untouchable till recently. We also try to get locally testable codes. If we will succeed this will be a real contribution to computer science: finding such codes is a major ( if not THE major) problem of modern error correcting codes. It represents a new paradigm: up to now all codes obtained were explicit constructions of what is known to exist by random consideration. Local testability is different: random codes are not locally testable!
We have established some of the goals, as well as some totally unexpected results which were not mentioned in the proposal, but are of fundamental importance. The main problem we still working on is the problem of locally testable codes.
We established: cut-off phenomena for Random walks on Ramanujan complexes, We constructed non-approximated groups. we proved first order rigidity for high rank arithmetic lattices.
Th emain thing we hope to do is finding locally testable codes.